Annette Witt

A comparative classification of complexity measures

Abstract A number of different measures of complexity have been described, discussed, and applied to the logistic map. A classification of these measures has been proposed, distinguishing homogeneous and generating partitions in phase space as well as structural and dynamical elements of the considered measure. The specific capabilities of particular measures to detect particular types of behavior of dynamical systems have been investigated and compared with each other.

Birth of strange nonchaotic attractors due to interior crisis

Abstract We study the interior crisis in the period-3-window of the quasiperiodically forced logistic map. Two routes from quasiperiodicity to chaos involving strange nonchaotic attractors (SNA) are discovered: Along one route we observe a sudden widening of the SNA. This is similar to the interior crisis in chaotic systems. Along the other route we find a direct transition from an invariant curve to a strange nonchaotic attractor exactly at the interior crisis point. This is a new mechanism of the appearance of strange nonchaotic attractors. Beyond the interior crisis the temporal behavior can be described as a crisis-induced intermittency, whose scaling behavior is discussed.

Testing stationarity in time series

We propose a procedure for testing stationarity of time series by combining a test for time independence of the probability density with one of the spectral density. The potentials of this test procedure are demonstrated by its application to various types of numerically simulated time series ranging from simple linear stochastic processes to high-dimensional transient chaos. Problems of practical implementation are discussed, in particular the relation between the lengths of the time series and its maximal relevant time scales. Stationarity is then tested for experimental data from geophysics and physiology. Exchange rates are found to be stationary on time scales of decades in the sense that their spectral densities do not significantly change.

TEMPORARY DIMENSIONS OF MULTIVARIATE DATA FROM PALEOCLIMATE RECORDS — A NOVEL MEASURE FOR DYNAMIC CHARACTERIZATION OF LONG-TERM CLIMATE CHANGE

Standard techniques of nonlinear data analysis are difficult to apply to short or insufficiently resolved multivariate time series. We demonstrate that dimension estimation based on the decay of eigenvalues of the covariance matrix yields qualitatively robust characteristics even in the case of very short measurement series. The uncertainty of these characteristics is discussed for both: synthetic data sets and measurement data. We apply this approach to measurements of trace element abundances in a marine sediment core obtained at the East Antarctic coast and discuss the results from a paleoclimate point of view.

Non-linear data analysis and statistical techniques in solar radio astronomy

We have discussed some tools from nonlinear dynamics which may help to analyze transient phenomena, such as solar bursts. The structure function known from turbulence theory is an appropriate method to find out some scaling behavior of fluctuations in time. More generally, the wavelet analysis, which is some generalization of the power spectrum, exhibits information on the location as well as the size of hidden characteristic features. Applying both techniques to microwave bursts, we have found some scaling properties that refer to the existence of hierarchic time structures. This is in good accordance with the electric circuit model for describing the flare-particle energization process.

Spatiotemporal Distributions of Unstable Periodic Orbits in Noisy Coupled Chaotic Systems

Techniques for detecting encounters with unstable periodic orbits (UPOs) have been very successful in the analysis of noisy, experimental time series. We present here a technique for applying the topological recurrence method of UPO detection to spatially extended systems. This approach is tested on a network of diffusively coupled chaotic Rossler systems, with both symmetric and asymmetric coupling schemes. We demonstrate how to extract encounters with UPOs from such data, and present a preliminary method for analyzing the results and extracting dynamical information from the data, based on a linear correlation analysis of the spatiotemporal occurrence of encounters with these low period UPOs. This analysis can provide an insight into the coupling structure of such a spatially extended system.